Recovering Asymptotics of Short Range Potentials

被引:0
|
作者
M. S. Joshi
A. Sá Barreto
机构
[1] Department of Pure Mathematics and Mathematical Statistics,
[2] University of Cambridge,undefined
[3] 16 Mill Lane,undefined
[4] Cambridge CB2 1SB,undefined
[5] England,undefined
[6] UK. E-mail: joshi@dpmms.cam.ac.uk,undefined
[7] Department of Mathematics,undefined
[8] Purdue University,undefined
[9] West Lafayette IN 47907,undefined
[10] Indiana,undefined
[11] USA.¶E-mail: sabarre@math.purdue.edu,undefined
来源
Communications in Mathematical Physics | 1998年 / 193卷
关键词
Fourier; Manifold; Euclidean Space; Integral Operator; Taylor Series;
D O I
暂无
中图分类号
学科分类号
摘要
Any compact smooth manifold with boundary admits a Riemann metric of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} near the boundary, where x is the boundary defining function and h' restricts to a Riemannian metric, h, on the boundary. Melrose has associated a scattering matrix to such a metric which was shown by he and Zworski to be a Fourier integral operator. It is shown here that the principal symbol of the difference of the scattering matrices for two potentials \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} at fixed energy determines a weighted integral of the lead term of V1 - V2 over all geodesics on the boundary. This is used to prove that the entire Taylor series of the potential at the boundary is determined by the scattering matrix at a non-zero fixed energy for certain manifolds including Euclidean space.
引用
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页码:197 / 208
页数:11
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